tensor product of two column stochastic matrix

Question

Is the tensor product of two column/row stochastic matrix is again a column/row stochastic? Thanks for helping.

Answer 1

The tensor product of two $n\times n$ matrices $A,B$ is $$A\otimes B = \left(\begin{array}{ccc} a_{11}B & \ldots& a_{1n}B\\ \vdots & \ldots & \vdots\\ a_{n1}B & \ldots & a_{nn} B \end{array}\right).$$ So if $A,B$ are column stochastic, then so is $A\otimes B$. Just check the $j$-th column of the block matrix $A\otimes B$ and take inside the block matrix the $k$-th column of $B$: $$a_{1j}[b_{1k}+\ldots+b_{nk}] + \ldots + a_{nj}[b_{1k}+\ldots+b_{nk}] = a_{1j}+\ldots+a_{nj} = 1.$$ Similar for the rows.